Variance over Z and moments of L-functions

Abstract: One of the central problems in analytic number theory has been to evaluate moments of the absolute value of L-functions on the critical line. Bounds on these moments are approximations to the Lindelöf hypothesis and, thus, subconvexity bounds for these L-functions. Besides a few low moments where rigorous results are known, sharp bounds on higher moments are wide open. Recently, in 2018, it has been discovered that there is a certain connection between asymptotics of moments of L-functions and variance over the integers (the Keating--Rodgers--Roditty-Gershon--Rudnick--Soundararajan conjecture in arithmetic progressions). Certain analogues of this conjecture are completely known, i.e., are theorems, in the function field setting. In this lecture, I plan to explain this new connection between asymptotics of variance over Z and those of moments, and discuss my work on confirming a smoothed version of this conjecture in a restricted range.

number theory

Audience: researchers in the topic

Carleton-Ottawa Number Theory seminar